WEBVTT 00:00:00.070 --> 00:00:02.410 The following content is provided under a Creative 00:00:02.410 --> 00:00:03.830 Commons license. 00:00:03.830 --> 00:00:06.040 Your support will help MIT OpenCourseWare 00:00:06.040 --> 00:00:10.130 continue to offer high-quality educational resources for free. 00:00:10.130 --> 00:00:12.670 To make a donation or to view additional materials 00:00:12.670 --> 00:00:16.630 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.630 --> 00:00:17.503 at ocw.mit.edu. 00:00:21.890 --> 00:00:27.440 ROBERT FIELD: Well, we're now well underway into quantum 00:00:27.440 --> 00:00:30.590 mechanics. 00:00:30.590 --> 00:00:38.510 So a lot of the important stuff goes by very fast. 00:00:38.510 --> 00:00:42.950 So we represent a quantum mechanical operator 00:00:42.950 --> 00:00:46.070 with a little hat, and it means do something 00:00:46.070 --> 00:00:48.720 to the thing on its right. 00:00:48.720 --> 00:00:50.520 And it has to be a linear operator, 00:00:50.520 --> 00:00:53.670 and you want to be sure you know what a linear operator does 00:00:53.670 --> 00:00:56.370 and what is not a linear operator. 00:00:56.370 --> 00:00:58.630 This is an eigenvalue equation. 00:00:58.630 --> 00:01:00.780 So we have some function which, when 00:01:00.780 --> 00:01:03.120 the operator operates on it, gives back 00:01:03.120 --> 00:01:05.880 a constant times that function. 00:01:05.880 --> 00:01:09.120 The constant is the eigenvalue, and the function 00:01:09.120 --> 00:01:12.180 is an eigenfunction of the operator that 00:01:12.180 --> 00:01:16.860 belongs to this eigenvalue, and all of quantum mechanics 00:01:16.860 --> 00:01:20.520 can be expressed in terms of eigenvalue equations. 00:01:20.520 --> 00:01:23.670 It's very important, and you sort of take it for granted. 00:01:23.670 --> 00:01:28.950 Now, one of the important things about quantum mechanics 00:01:28.950 --> 00:01:33.600 is that we have to find a linear operator that 00:01:33.600 --> 00:01:37.090 corresponds to the classically observable quantities. 00:01:37.090 --> 00:01:40.050 And for x the linear operator is x, 00:01:40.050 --> 00:01:42.210 and for the momentum the linear operator 00:01:42.210 --> 00:01:45.150 is minus ih bar partial with respect to x. 00:01:45.150 --> 00:01:47.260 That should bother you two ways. 00:01:47.260 --> 00:01:51.600 One is the i, and the other is the partial derivative. 00:01:51.600 --> 00:01:55.380 But when you apply this operator to functions, 00:01:55.380 --> 00:02:02.070 you discover that out pops something that has the expected 00:02:02.070 --> 00:02:04.920 behavior of momentum, and so this 00:02:04.920 --> 00:02:07.590 is in fact the operator that we're 00:02:07.590 --> 00:02:10.710 going to use for momentum. 00:02:10.710 --> 00:02:14.550 And then there is a commutation rule. 00:02:14.550 --> 00:02:22.140 This commutation rule, xp minus px, is equal to this. 00:02:22.140 --> 00:02:27.380 This is really the foundation of quantum mechanics, 00:02:27.380 --> 00:02:31.900 and as I've said before, many people derive everything 00:02:31.900 --> 00:02:34.580 from a few commutation rules. 00:02:34.580 --> 00:02:38.140 It's really scary, but you should 00:02:38.140 --> 00:02:41.890 be able to work out this commutation 00:02:41.890 --> 00:02:47.950 rule by applying xp minus px to some arbitrary function. 00:02:47.950 --> 00:02:51.160 And going through the symbolics should take you about 30 00:02:51.160 --> 00:02:53.380 seconds, or maybe it shouldn't. 00:02:53.380 --> 00:02:56.360 Maybe you can go faster. 00:02:56.360 --> 00:02:56.860 OK. 00:03:00.020 --> 00:03:03.060 We have an operator, and we often 00:03:03.060 --> 00:03:06.660 want to know what is the expectation 00:03:06.660 --> 00:03:09.840 value of the particular function, which 00:03:09.840 --> 00:03:13.440 we could symbolize here, but it's never done that way. 00:03:13.440 --> 00:03:16.500 So we have some function, and we want 00:03:16.500 --> 00:03:19.500 to calculate its expectation value of operator A, 00:03:19.500 --> 00:03:22.870 and this is it. 00:03:22.870 --> 00:03:26.860 And so this is a normalization integral, 00:03:26.860 --> 00:03:31.330 and this normalization integral is usually taken for granted, 00:03:31.330 --> 00:03:36.520 because we almost always work with sets of functions 00:03:36.520 --> 00:03:38.620 which are normalized. 00:03:38.620 --> 00:03:41.800 And so if you convince yourself that it is, 00:03:41.800 --> 00:03:45.710 in fact, they are normalized, fine, 00:03:45.710 --> 00:03:48.940 and then this is the thing that you normally would calculate. 00:03:53.840 --> 00:03:56.810 Then, we went to our first problem 00:03:56.810 --> 00:04:00.230 in quantum mechanics which is the free particle, 00:04:00.230 --> 00:04:05.480 and the free particle has some idiosyncrasies. 00:04:05.480 --> 00:04:09.780 The wave function for the free particle 00:04:09.780 --> 00:04:18.370 has the form e to the ikx plus e to the minus ikx, 00:04:18.370 --> 00:04:27.270 and the Hamiltonian is minus h bar 00:04:27.270 --> 00:04:36.900 squared over 2m second partial with respect to x plus v0. 00:04:36.900 --> 00:04:48.130 So there's no v0 here, and we have two different exponentials 00:04:48.130 --> 00:04:53.680 and so is this really going to be an eigenfunction 00:04:53.680 --> 00:04:54.970 of the Hamiltonian? 00:04:54.970 --> 00:05:01.060 This is really p squared, and so this 00:05:01.060 --> 00:05:03.280 is going to be an eigenfunction of p squared too. 00:05:08.950 --> 00:05:11.450 All right, so let's show a little picture here. 00:05:11.450 --> 00:05:19.530 Here is energy, and this is v0, and let's say 00:05:19.530 --> 00:05:20.850 this is the 0 of energy. 00:05:43.040 --> 00:05:46.900 What are the eigenvalues of this? 00:05:46.900 --> 00:05:51.290 What does the Hamiltonian do to this function? 00:05:55.050 --> 00:06:01.310 Well, in order to do that, you have to calculate something 00:06:01.310 --> 00:06:06.320 like where you have to calculate the second derivative of each 00:06:06.320 --> 00:06:09.650 of these terms, and the second derivative of this term 00:06:09.650 --> 00:06:13.805 brings down a minus k squared. 00:06:13.805 --> 00:06:16.240 Now, the second derivative of this term 00:06:16.240 --> 00:06:22.970 brings down a minus k squared, and so the energy eigenvalues 00:06:22.970 --> 00:06:32.270 are going to be given by h bar squared a squared over 2m plus 00:06:32.270 --> 00:06:33.720 v0. 00:06:33.720 --> 00:06:38.430 So these are the eigenvalues, eigenfunctions, 00:06:38.430 --> 00:06:44.870 the energies of a free particle, and they're not quantized. 00:06:44.870 --> 00:06:48.780 Now this v0 is something that will often trip you up, 00:06:48.780 --> 00:06:52.630 because it's hidden here. 00:06:52.630 --> 00:06:53.340 It's not in here. 00:06:55.770 --> 00:06:56.270 OK. 00:07:00.300 --> 00:07:03.030 I'm going to torture you with something. 00:07:03.030 --> 00:07:10.000 So why are these two k's the same? 00:07:10.000 --> 00:07:14.990 What would happen if the k for the positive term 00:07:14.990 --> 00:07:17.080 were different from the k for the negative term? 00:07:19.720 --> 00:07:22.051 Simple answer. 00:07:22.051 --> 00:07:22.550 Yeah? 00:07:22.550 --> 00:07:24.476 AUDIENCE: There'd be two eigenvalues? 00:07:24.476 --> 00:07:25.600 ROBERT FIELD: That's right. 00:07:25.600 --> 00:07:29.355 It wouldn't be an eigenvalue. 00:07:29.355 --> 00:07:31.480 It wouldn't be an eigenfunction of the Hamiltonian. 00:07:31.480 --> 00:07:33.920 It's a mixture of two eigenvalues, 00:07:33.920 --> 00:07:36.670 and so that's simple. 00:07:36.670 --> 00:07:42.430 But often we might be dealing with a potential that's 00:07:42.430 --> 00:07:47.440 not simple like this but has got complexity. 00:07:50.500 --> 00:07:54.605 So suppose we had a potential that did this. 00:07:59.270 --> 00:08:05.380 The potential is constant piecewise, 00:08:05.380 --> 00:08:06.270 and so what do we do? 00:08:09.311 --> 00:08:09.810 Yes? 00:08:09.810 --> 00:08:11.610 AUDIENCE: Break down the function into pieces 00:08:11.610 --> 00:08:12.901 for each in certain boundaries? 00:08:12.901 --> 00:08:16.710 ROBERT FIELD: Yes, and that's exactly right. 00:08:16.710 --> 00:08:19.700 You do want to break it up, but one 00:08:19.700 --> 00:08:21.800 of the things I'm stressing here is 00:08:21.800 --> 00:08:24.110 that you want to be able to draw cartoons. 00:08:26.800 --> 00:08:32.820 And so we know that if we choose an energy here, 00:08:32.820 --> 00:08:39.860 there is a certain momentum, or a certain kinetic energy here, 00:08:39.860 --> 00:08:45.280 and a different kinetic energy here, and so somehow, 00:08:45.280 --> 00:08:50.680 what you write for the wave function will reflect that. 00:08:50.680 --> 00:08:53.650 But now qualitatively, pictorially, 00:08:53.650 --> 00:08:59.830 if we have a wave function in this region which 00:08:59.830 --> 00:09:03.010 is oscillating like this, and it'll 00:09:03.010 --> 00:09:06.790 be oscillating at the same spatial frequency over here, 00:09:06.790 --> 00:09:08.930 well what's going to be happening here? 00:09:08.930 --> 00:09:10.930 Is it going to be oscillating faster or slower? 00:09:14.157 --> 00:09:16.605 AUDIENCE: Faster. 00:09:16.605 --> 00:09:18.520 Faster. 00:09:18.520 --> 00:09:21.120 ROBERT FIELD: Absolutely, and is it 00:09:21.120 --> 00:09:25.325 going to have amplitude smaller or larger than here? 00:09:29.600 --> 00:09:31.170 You're going to answer, yes? 00:09:33.740 --> 00:09:38.040 Well, let me do a thought experiment. 00:09:38.040 --> 00:09:41.240 So I'm going to walk from one side of the blackboard 00:09:41.240 --> 00:09:44.680 to the other, and I'm going to walk at a constant velocity. 00:09:44.680 --> 00:09:47.705 Then, I'm going to walk faster and then back 00:09:47.705 --> 00:09:51.310 to this original velocity. 00:09:51.310 --> 00:09:54.040 So what's the probability of seeing me 00:09:54.040 --> 00:09:58.750 in the middle region relative to the edge regions? 00:09:58.750 --> 00:10:00.800 Yes? 00:10:00.800 --> 00:10:02.646 AUDIENCE: It's less. 00:10:02.646 --> 00:10:03.770 ROBERT FIELD: That's right. 00:10:03.770 --> 00:10:06.650 The probability, local probability, 00:10:06.650 --> 00:10:10.330 is proportional to 1 over the velocity, 00:10:10.330 --> 00:10:12.980 and the wave function is proportional to 1 00:10:12.980 --> 00:10:15.890 over the square root of the velocity. 00:10:15.890 --> 00:10:17.970 And the velocity is related to the momentum, 00:10:17.970 --> 00:10:19.680 and so we have everything. 00:10:19.680 --> 00:10:26.070 So we know that the wave here will be oscillating faster 00:10:26.070 --> 00:10:28.740 and with lower amplitude. 00:10:28.740 --> 00:10:32.220 This is what I want you to know, and you'll 00:10:32.220 --> 00:10:34.860 be able to use that cartoon to solve problems. 00:10:37.420 --> 00:10:40.580 If you understand what's going on here, 00:10:40.580 --> 00:10:43.090 these pictures will be equivalent 00:10:43.090 --> 00:10:46.720 to global understanding, and these pictures 00:10:46.720 --> 00:10:57.480 are also part of semi-classical quantum mechanics. 00:10:57.480 --> 00:11:01.980 I believe you all know classical mechanics at least a little, 00:11:01.980 --> 00:11:03.960 enough to be useful. 00:11:03.960 --> 00:11:07.220 And what we want to be able to do in order to draw pictures 00:11:07.220 --> 00:11:11.130 and to understand stuff is to insert just enough quantum 00:11:11.130 --> 00:11:14.870 mechanics into classical mechanics so that it's correct. 00:11:17.640 --> 00:11:21.650 Then, all of a sudden, it starts to make a lot more sense. 00:11:21.650 --> 00:11:22.150 OK. 00:11:36.110 --> 00:11:44.190 So the particle in a box, well, we have this sort of situation, 00:11:44.190 --> 00:11:49.830 and we have 0 and a. 00:11:49.830 --> 00:11:56.730 So the length of the box is a, and the bottom of the box v0 00:11:56.730 --> 00:12:00.392 is 0 for this picture. 00:12:00.392 --> 00:12:02.600 Now, one of the things that I want you to think about 00:12:02.600 --> 00:12:05.100 is, OK, I understand. 00:12:05.100 --> 00:12:06.920 I've solved this problem. 00:12:06.920 --> 00:12:08.870 I know how to solve this problem. 00:12:08.870 --> 00:12:10.762 I know how to get the eigenvalues, 00:12:10.762 --> 00:12:12.470 and I know how to get the eigenfunctions, 00:12:12.470 --> 00:12:14.970 and I know how to normalize them. 00:12:14.970 --> 00:12:19.550 Well, suppose I move the box to the side. 00:12:19.550 --> 00:12:25.970 So I move it from say b to a plus b. 00:12:25.970 --> 00:12:29.406 So it's the same width, but it's just in a different place. 00:12:29.406 --> 00:12:30.630 Well, did anything change? 00:12:37.830 --> 00:12:39.960 The only thing that changes is the wave function, 00:12:39.960 --> 00:12:43.690 because you have to shift the coordinates. 00:12:43.690 --> 00:12:47.570 What happens if I raise the box or lower the box? 00:12:47.570 --> 00:12:49.861 Will anything change? 00:12:49.861 --> 00:12:51.690 AUDIENCE: [INAUDIBLE] 00:12:51.690 --> 00:12:52.850 ROBERT FIELD: You're hot. 00:12:52.850 --> 00:12:55.010 AUDIENCE: [INAUDIBLE] 00:12:55.256 --> 00:12:56.256 ROBERT FIELD: I'm sorry? 00:12:56.256 --> 00:12:58.050 AUDIENCE: [INAUDIBLE] 00:13:00.119 --> 00:13:00.910 ROBERT FIELD: Yeah. 00:13:00.910 --> 00:13:06.680 If I move the box so that v0 is not 0, but v0 is 10. 00:13:06.680 --> 00:13:10.490 AUDIENCE: Then, the weight function will oscillate slower. 00:13:10.490 --> 00:13:12.095 ROBERT FIELD: No. 00:13:12.095 --> 00:13:14.280 AUDIENCE: [INAUDIBLE] 00:13:14.280 --> 00:13:20.600 ROBERT FIELD: So if you move the box up in energy, 00:13:20.600 --> 00:13:24.321 the wave function is going to look exactly the same, 00:13:24.321 --> 00:13:26.820 but the energies are going to be different by the amount you 00:13:26.820 --> 00:13:30.840 move the box up or down, and this is really important. 00:13:30.840 --> 00:13:35.580 It may seem trivial to some of you and really obscure 00:13:35.580 --> 00:13:38.670 to others, but you really want to be 00:13:38.670 --> 00:13:41.370 able to take these things apart. 00:13:41.370 --> 00:13:44.070 Because that will enable you to understand them 00:13:44.070 --> 00:13:49.400 in a permanent way, and the cartoons are really important. 00:13:49.400 --> 00:13:53.480 So if you have the solution to the particle in a box, 00:13:53.480 --> 00:13:56.010 then it doesn't matter where the box is. 00:13:56.010 --> 00:13:58.515 You know the solution to any particle in a box. 00:14:04.950 --> 00:14:05.940 OK. 00:14:05.940 --> 00:14:10.560 There is something that I meant to talk about briefly, 00:14:10.560 --> 00:14:18.350 but when we write these solutions-- where 00:14:18.350 --> 00:14:22.210 did the other blackboard go? 00:14:22.210 --> 00:14:25.220 All right, well, I've hidden it-- 00:14:25.220 --> 00:14:29.460 so when we have solutions like e to the ikx 00:14:29.460 --> 00:14:36.920 and e to the minus ikx, so we have say a here and b here. 00:14:36.920 --> 00:14:39.903 When we go to normalize a function like this-- 00:14:39.903 --> 00:14:42.050 let's put the plus in here-- 00:14:42.050 --> 00:14:49.220 then we write psi star psi dx. 00:14:49.220 --> 00:14:52.430 So psi star would make this go a star 00:14:52.430 --> 00:14:55.160 and this go to e to the minus ikx, 00:14:55.160 --> 00:15:00.960 and this go to b star e to the plus ikx. 00:15:00.960 --> 00:15:02.870 So now, we multiply things together. 00:15:02.870 --> 00:15:06.530 We get an a, a star which is the square modulus of a, 00:15:06.530 --> 00:15:09.680 and we get e to the ikx and e to the minus ikx. 00:15:09.680 --> 00:15:11.480 It's 1. 00:15:11.480 --> 00:15:13.780 This is why we use this form. 00:15:13.780 --> 00:15:18.560 The integrals for things involving e to the ikx 00:15:18.560 --> 00:15:20.850 are either 1 or 0. 00:15:23.530 --> 00:15:28.650 So if you took e to the ikx, this term, 00:15:28.650 --> 00:15:30.780 and multiplied it by this term, you'd get an a, 00:15:30.780 --> 00:15:35.330 b star e to the 2 ikx integrated over a finite region. 00:15:35.330 --> 00:15:35.850 That's 0. 00:15:38.880 --> 00:15:42.060 So we really like this exponential notation, 00:15:42.060 --> 00:15:45.180 even if you've been brought up on sines and cosines, 00:15:45.180 --> 00:15:47.880 and you use the sines and cosines to impose the boundary 00:15:47.880 --> 00:15:48.995 conditions. 00:16:01.890 --> 00:16:06.670 OK, another challenge. 00:16:06.670 --> 00:16:16.230 So this is v0, and the only problem is this v0-- 00:16:16.230 --> 00:16:21.310 well, it looks like this. 00:16:21.310 --> 00:16:25.760 So this is v1. 00:16:25.760 --> 00:16:31.880 OK, so we have now a particle in this straight. 00:16:31.880 --> 00:16:35.570 It's a hybrid between the free particle 00:16:35.570 --> 00:16:36.605 and a particle in a box. 00:16:40.290 --> 00:16:42.150 So suppose we're at an energy like this. 00:16:45.000 --> 00:16:47.160 What's going to happen? 00:16:47.160 --> 00:16:50.610 Well, everything that's outside-- 00:16:50.610 --> 00:16:54.120 everything that's in the classically-allowed region, 00:16:54.120 --> 00:16:54.990 we understand. 00:16:54.990 --> 00:17:00.860 We know how to deal with it, but in here, well, that's OK too. 00:17:00.860 --> 00:17:04.819 But inside this classically-forbidden region, 00:17:04.819 --> 00:17:06.980 the wave function is going to behave differently. 00:17:06.980 --> 00:17:11.280 Now, I'm going to assert something. 00:17:11.280 --> 00:17:12.900 It doesn't have nodes. 00:17:12.900 --> 00:17:14.520 It doesn't oscillate. 00:17:14.520 --> 00:17:16.319 It's either exponentially decreasing 00:17:16.319 --> 00:17:24.750 or exponentially increasing, and it will never cross 0, never. 00:17:24.750 --> 00:17:25.670 OK. 00:17:25.670 --> 00:17:32.060 So now, if we're solving a problem involving 00:17:32.060 --> 00:17:42.300 any kind of 1D potential, number of nodes. 00:17:58.610 --> 00:18:07.190 So for 2D-bound problems, the number of nodes starts with 0, 00:18:07.190 --> 00:18:10.070 and it corresponds to the lowest energy state. 00:18:10.070 --> 00:18:15.450 The next state up has 1 node, and the next state has 2 nodes. 00:18:15.450 --> 00:18:16.940 So by counting the nodes, you would 00:18:16.940 --> 00:18:21.890 know what the energy order is of these eigenvalues which is also 00:18:21.890 --> 00:18:24.080 an extremely useful thing. 00:18:24.080 --> 00:18:26.420 If you're thinking about it or telling your computer 00:18:26.420 --> 00:18:29.750 to find the 33rd eigenvalue of something, 00:18:29.750 --> 00:18:34.550 because you just run a calculation that solves 00:18:34.550 --> 00:18:38.090 for an approximate wave function, and the 33rd, 00:18:38.090 --> 00:18:43.040 it needs 32 nodes. 00:18:43.040 --> 00:18:47.150 And so the computer says, oh, thank you, master, 00:18:47.150 --> 00:18:49.572 and here is your wave function, but you 00:18:49.572 --> 00:18:50.780 have to find the right thing. 00:18:50.780 --> 00:18:51.500 OK. 00:18:51.500 --> 00:19:01.040 Now, here is the picture that you use to remember everything 00:19:01.040 --> 00:19:02.240 about a particle in a box. 00:19:05.610 --> 00:19:08.030 And the wave function looks like this, 00:19:08.030 --> 00:19:10.565 and the next wave function looks like that, 00:19:10.565 --> 00:19:16.270 and the next wave function looks like this. 00:19:16.270 --> 00:19:24.280 And so no nodes, 1 node, 2 nodes, the nodes 00:19:24.280 --> 00:19:29.530 are symmetrically arranged in the space available. 00:19:29.530 --> 00:19:34.120 And the lobes on one side of the node and the other side 00:19:34.120 --> 00:19:38.360 have the same amplitude, different sine, 00:19:38.360 --> 00:19:40.550 and they're all normalized. 00:19:40.550 --> 00:19:43.790 And so the maximum value for each of these guys 00:19:43.790 --> 00:19:48.920 is 2 over a square root, where this is 0 to a. 00:19:51.950 --> 00:19:54.570 So that's a fantastic simplification, 00:19:54.570 --> 00:19:59.060 and it also reminds you of Mr. DeBroglie. 00:19:59.060 --> 00:20:01.040 He said, you have to have an integer 00:20:01.040 --> 00:20:03.770 number of half wavelengths-- 00:20:03.770 --> 00:20:06.140 well, for the hydrogen-- an integer number 00:20:06.140 --> 00:20:09.150 of wavelengths around a path. 00:20:09.150 --> 00:20:11.220 And for here, you need an integer number 00:20:11.220 --> 00:20:12.910 of wavelengths for that round trip 00:20:12.910 --> 00:20:14.760 which is the same thing or an integer 00:20:14.760 --> 00:20:18.880 number of half wavelengths. 00:20:18.880 --> 00:20:22.850 That's DeBroglie's idea, and it enables 00:20:22.850 --> 00:20:25.580 you to say, oh well, let's see if we 00:20:25.580 --> 00:20:28.640 can use this concept of wavelength 00:20:28.640 --> 00:20:30.530 to approach general problems. 00:20:38.650 --> 00:20:39.160 OK. 00:20:39.160 --> 00:20:45.520 Well, if you do something to the potential 00:20:45.520 --> 00:20:52.010 by putting a little thing in it, well, the wave function 00:20:52.010 --> 00:20:55.660 will oscillate more slowly in that region, 00:20:55.660 --> 00:21:02.960 and that causes it to be at a higher or lower energy? 00:21:02.960 --> 00:21:05.520 If it's oscillating more slowly here, 00:21:05.520 --> 00:21:09.740 it has to make it to an integer number of half wavelengths, 00:21:09.740 --> 00:21:12.170 and so that means it pushes it up. 00:21:12.170 --> 00:21:16.120 And if you do this, it'll push it down, 00:21:16.120 --> 00:21:18.490 and you can do terrible things. 00:21:18.490 --> 00:21:21.280 You can put a delta function there, 00:21:21.280 --> 00:21:24.370 and now you know everything qualitatively 00:21:24.370 --> 00:21:26.065 that can happen in a 1D box. 00:21:30.661 --> 00:21:31.160 OK. 00:21:31.160 --> 00:21:33.620 One of the things that bothers people a lot 00:21:33.620 --> 00:21:37.930 is, OK, so we have some wave function, 00:21:37.930 --> 00:21:42.590 it's got lots of nodes, and the particle starts out over here. 00:21:42.590 --> 00:21:45.006 How did it get across the node? 00:21:45.006 --> 00:21:46.380 How does it move across the node? 00:21:48.980 --> 00:21:51.740 Well, the answer is it's not moving. 00:21:51.740 --> 00:21:52.300 It's here. 00:21:52.300 --> 00:21:52.690 It's here. 00:21:52.690 --> 00:21:53.189 It's here. 00:21:53.189 --> 00:21:56.340 It's everywhere, and this is just the probability amplitude. 00:21:56.340 --> 00:22:02.200 There is no motion through a node, no motion at all. 00:22:02.200 --> 00:22:04.000 We are going to do time-dependent quantum 00:22:04.000 --> 00:22:07.150 mechanics before too long, and then there will be motion, 00:22:07.150 --> 00:22:11.710 but that motion is encoded in a different way. 00:22:14.220 --> 00:22:14.720 OK. 00:22:14.720 --> 00:22:21.110 Another thing, suppose you have a particle in a box, 00:22:21.110 --> 00:22:24.320 and it's in some state, and I'm going to draw something 00:22:24.320 --> 00:22:26.750 like this again. 00:22:26.750 --> 00:22:29.300 OK, first of all, one, two, three, four, five, 00:22:29.300 --> 00:22:31.730 which state is that? 00:22:39.010 --> 00:22:43.081 I got-- the hands are right, six, it's the sixth eigenstate. 00:22:43.081 --> 00:22:43.580 OK. 00:22:43.580 --> 00:22:45.770 Now, suppose-- nothing is moving. 00:22:45.770 --> 00:22:46.610 Right? 00:22:46.610 --> 00:22:49.250 This is a stationary state. 00:22:49.250 --> 00:22:52.940 How would you experimentally, in principle, 00:22:52.940 --> 00:23:00.710 determine that the particle is in this n equals 6 state? 00:23:03.840 --> 00:23:07.530 Now, this can be a completely fanciful experiment, which 00:23:07.530 --> 00:23:10.740 you would never do, but you could still 00:23:10.740 --> 00:23:14.490 describe how you would do it and what it would tell you. 00:23:14.490 --> 00:23:17.420 And so, yes. 00:23:17.420 --> 00:23:19.580 AUDIENCE: Try to find the n equals 6 to n 00:23:19.580 --> 00:23:22.802 equals 7 transition by irradiating it or something? 00:23:22.802 --> 00:23:23.510 ROBERT FIELD: OK. 00:23:23.510 --> 00:23:27.260 That's the quantum mechanical-- 00:23:27.260 --> 00:23:30.320 I agree, spectroscopy wins always. 00:23:30.320 --> 00:23:37.780 But if you want to observe the wave function or something 00:23:37.780 --> 00:23:42.460 related to the wave function, like the number of nodes, 00:23:42.460 --> 00:23:44.270 what would you do? 00:23:44.270 --> 00:23:48.310 And the reason I'm being very apologetic about this 00:23:48.310 --> 00:23:50.830 is because it's a crazy idea, But this 00:23:50.830 --> 00:23:52.575 is a one-dimensional system. 00:23:52.575 --> 00:23:53.075 Right? 00:23:53.075 --> 00:23:57.260 It's in the blackboard, and so you could stand out here 00:23:57.260 --> 00:24:01.580 and shoot particles at it from the perpendicular direction 00:24:01.580 --> 00:24:04.070 and collect the number of times you have a hit. 00:24:07.600 --> 00:24:11.020 And so you would discover that you 00:24:11.020 --> 00:24:16.180 would measure a probability distribution which 00:24:16.180 --> 00:24:19.465 had the form-- 00:24:25.060 --> 00:24:27.720 well, I can't draw this properly. 00:24:27.720 --> 00:24:31.380 It's going to have one, two, three, four, five, six, 00:24:31.380 --> 00:24:37.200 six regions separated by a gap, and what it's measuring 00:24:37.200 --> 00:24:39.500 is psi 6 squared. 00:24:42.020 --> 00:24:47.930 Well, you can't measure, you cannot observe a wave function, 00:24:47.930 --> 00:24:51.850 but you can observe a probability distribution wave 00:24:51.850 --> 00:24:52.750 function squared. 00:24:55.520 --> 00:24:57.840 You can also do a spectroscopic experiment 00:24:57.840 --> 00:25:01.184 and find out what is the nature of the Hamiltonian. 00:25:01.184 --> 00:25:03.100 And if you know the nature of the Hamiltonian, 00:25:03.100 --> 00:25:07.039 you can calculate the wave function, 00:25:07.039 --> 00:25:08.080 but you can't observe it. 00:25:13.830 --> 00:25:14.330 OK. 00:25:14.330 --> 00:25:19.330 Another thing, this harmonic oscillator-- this particle 00:25:19.330 --> 00:25:22.784 in a box has a minimum energy which is not 00:25:22.784 --> 00:25:23.825 at the bottom of the box. 00:25:33.530 --> 00:25:36.355 Well, we have something called the uncertainty principle. 00:25:42.300 --> 00:25:44.980 Now, I'm just pulling this out of my pocket, 00:25:44.980 --> 00:25:52.450 but I know that x, p is equal to i h bar not 0, 00:25:52.450 --> 00:25:57.880 and one can derive some uncertainty principle by doing 00:25:57.880 --> 00:25:59.830 a little bit more mathematics. 00:25:59.830 --> 00:26:02.860 But basically that uncertainty principle 00:26:02.860 --> 00:26:20.350 is where sigma x is expectation value 00:26:20.350 --> 00:26:27.270 of x squared minus expectation value of x squared square root. 00:26:27.270 --> 00:26:29.800 So if we can calculate this and calculate that, 00:26:29.800 --> 00:26:33.750 we can calculate the variance in x, 00:26:33.750 --> 00:26:35.600 and you can calculate the variance in p. 00:26:35.600 --> 00:26:38.680 That's exact, and that's what you 00:26:38.680 --> 00:26:48.130 can derive from the computation rule, but for our purposes, 00:26:48.130 --> 00:26:49.510 we can be really crude. 00:26:49.510 --> 00:26:55.400 And so if I'm in this state, what is delta x? 00:26:59.590 --> 00:27:02.752 What is the range of possibilities for x? 00:27:02.752 --> 00:27:04.210 AUDIENCE: The box link? 00:27:04.210 --> 00:27:06.700 ROBERT FIELD: Yeah, a. 00:27:06.700 --> 00:27:12.800 OK, and what is the possibility-- 00:27:12.800 --> 00:27:15.640 what is the uncertainty in p sub x? 00:27:24.000 --> 00:27:29.340 In an eigenstate, we've got equal amplitudes going this way 00:27:29.340 --> 00:27:30.180 and going that way. 00:27:33.770 --> 00:27:44.510 So we could just say p sub x positive minus p sub x negative 00:27:44.510 --> 00:27:45.485 which is 2p. 00:27:48.550 --> 00:27:51.490 That's the uncertainty. 00:27:51.490 --> 00:27:55.120 And if we know what quantum number we're in we 00:27:55.120 --> 00:28:00.670 know what the expectation value for p of the momentum is, 00:28:00.670 --> 00:28:10.810 and what we derive is that delta x delta P is equal to hn. 00:28:15.290 --> 00:28:18.490 You can do that, and maybe I should ask you to really 00:28:18.490 --> 00:28:19.490 be sure you can do that. 00:28:25.250 --> 00:28:29.620 In seconds, because you really know 00:28:29.620 --> 00:28:35.020 what the possible values of momentum or momentum squared 00:28:35.020 --> 00:28:36.775 are for a product of a box. 00:28:39.811 --> 00:28:40.310 OK. 00:28:43.190 --> 00:28:47.690 So why is there zero-point energy, because if you said, 00:28:47.690 --> 00:28:51.140 I had a level at the bottom of the box, 00:28:51.140 --> 00:28:55.170 we would have the momentum 0. 00:28:55.170 --> 00:28:58.370 The uncertainty and the momentum is 0, 00:28:58.370 --> 00:29:00.067 and the product of the uncertainty 00:29:00.067 --> 00:29:02.150 of the moment times the product of the uncertainty 00:29:02.150 --> 00:29:06.540 in the position has to be some finite number, 00:29:06.540 --> 00:29:08.580 and you can't do that here. 00:29:08.580 --> 00:29:11.430 And so this is a simple illustration of the uncertainty 00:29:11.430 --> 00:29:16.820 principle that you have to have a non-zero zero-point energy. 00:29:16.820 --> 00:29:18.770 That's true for all one-dimensional problems. 00:29:23.290 --> 00:29:24.096 OK. 00:29:24.096 --> 00:29:25.680 We've got lots of time. 00:29:33.830 --> 00:29:37.620 One of the beautiful things about quantum mechanics 00:29:37.620 --> 00:29:41.990 is that if you solved one problem, 00:29:41.990 --> 00:29:44.110 you could solve a whole bunch of problems, 00:29:44.110 --> 00:29:47.950 and so to illustrate that, let's consider 00:29:47.950 --> 00:29:50.893 the 3D particle in a box. 00:29:53.660 --> 00:29:58.760 So for the 3D particle in a box, the Hamiltonian 00:29:58.760 --> 00:30:03.050 can be written as a little Hamiltonian for the x degree 00:30:03.050 --> 00:30:06.960 of freedom y and z. 00:30:06.960 --> 00:30:08.270 OK. 00:30:08.270 --> 00:30:12.950 So we have three independent motions of the particle. 00:30:12.950 --> 00:30:15.590 They're not coupled. 00:30:15.590 --> 00:30:18.604 They could be, and we're interested in letting 00:30:18.604 --> 00:30:19.270 them be coupled. 00:30:19.270 --> 00:30:24.210 But that's where we start asking questions about reality, 00:30:24.210 --> 00:30:27.120 and that's where we bring in perturbation theory. 00:30:27.120 --> 00:30:30.270 But for this, oh, that's fantastic, 00:30:30.270 --> 00:30:33.460 because I know the eigenvalues of this operator 00:30:33.460 --> 00:30:36.290 and of this operator, eigenfunctions, 00:30:36.290 --> 00:30:37.800 and of this operator. 00:30:37.800 --> 00:30:39.480 So the problem is basically solved 00:30:39.480 --> 00:30:42.840 once you solve the 1D box. 00:30:42.840 --> 00:30:49.240 Now, one proviso, what you can do this separation completely 00:30:49.240 --> 00:30:59.140 formally as long as hx, hy commute, 00:30:59.140 --> 00:31:02.000 and basically we say the x, y, and z directions 00:31:02.000 --> 00:31:06.740 don't interact with each other. 00:31:06.740 --> 00:31:08.940 The particle is free inside the box. 00:31:08.940 --> 00:31:10.520 It's just encountering walls. 00:31:10.520 --> 00:31:16.490 There are no springs or anything expressing the number 00:31:16.490 --> 00:31:17.960 of degrees of freedom. 00:31:17.960 --> 00:31:19.670 OK. 00:31:19.670 --> 00:31:24.380 So we have now a wave function, which is a function of x, y, 00:31:24.380 --> 00:31:26.750 and z, but we can always write it 00:31:26.750 --> 00:31:36.890 as psi x of x, psi y of y, psi z of z. 00:31:36.890 --> 00:31:41.490 So it's a product of three wave functions that we know, 00:31:41.490 --> 00:31:47.920 and the energy is going to be expressed 00:31:47.920 --> 00:32:25.400 as a function of three quantum numbers, where the box is 00:32:25.400 --> 00:32:26.840 edge lengths a, b, and c. 00:32:30.110 --> 00:32:32.850 You didn't see me looking at my notes. 00:32:32.850 --> 00:32:38.380 I'm just taking the solution to 1D box, and I'm multiplying it. 00:32:38.380 --> 00:32:43.970 And so now we have the particle in a 3D box, 00:32:43.970 --> 00:32:49.020 and this is where the ideal gas law comes from, but not 00:32:49.020 --> 00:32:51.090 in this course. 00:32:51.090 --> 00:32:53.480 So anyway, this is a simple thing, 00:32:53.480 --> 00:32:55.850 and the wave functions are simple as well, 00:32:55.850 --> 00:32:59.600 and you can do all these fantastic things. 00:33:05.740 --> 00:33:11.220 So there are many problems like a polyatomic molecule. 00:33:11.220 --> 00:33:15.010 In a polyatomic molecule, if you have n atoms, 00:33:15.010 --> 00:33:20.280 you have 3n minus 6 vibrational modes. 00:33:20.280 --> 00:33:23.250 You might ask, what is a vibrational mode? 00:33:23.250 --> 00:33:26.640 Well, are they're independent motions 00:33:26.640 --> 00:33:33.210 of the atoms that satisfy the harmonic oscillator 00:33:33.210 --> 00:33:37.410 Hamiltonian, which we'll come to next time. 00:33:37.410 --> 00:33:44.490 And so we have 3n minus 6 exactly solved problems 00:33:44.490 --> 00:33:48.540 all cohabiting in one Hamiltonian. 00:33:48.540 --> 00:33:51.300 And then we can say, oh yeah, we got these oscillators, 00:33:51.300 --> 00:33:53.990 and if I stretch a particular bond, 00:33:53.990 --> 00:33:58.680 it might affect the force constant for the bending. 00:33:58.680 --> 00:34:03.720 So we can introduce couplings between the oscillators, 00:34:03.720 --> 00:34:06.320 and in fact, that's what we do with perturbation theory. 00:34:06.320 --> 00:34:08.030 That's the whole purpose. 00:34:08.030 --> 00:34:12.110 And with that, we can describe both the spectrum 00:34:12.110 --> 00:34:16.370 and how the spectrum encodes the couplings between the modes. 00:34:16.370 --> 00:34:18.800 And also, we can describe what's called intramolecular 00:34:18.800 --> 00:34:21.500 vibrational redistribution, which 00:34:21.500 --> 00:34:24.210 happens when you have a very high density 00:34:24.210 --> 00:34:25.760 of vibrational state. 00:34:25.760 --> 00:34:29.210 Energy moves around, because all the modes are coupled, 00:34:29.210 --> 00:34:31.610 and so even if you've plucked one, 00:34:31.610 --> 00:34:33.320 the excitation would go to others. 00:34:33.320 --> 00:34:37.340 And we can understand that all using the same formalism 00:34:37.340 --> 00:34:39.170 that we're about to develop. 00:34:52.070 --> 00:34:54.670 All right. 00:34:54.670 --> 00:34:56.760 I'm not using my notes this time, 00:34:56.760 --> 00:34:59.600 because I think there's just so much insight, 00:34:59.600 --> 00:35:02.105 so I have to keep checking to see what I've skipped. 00:35:06.881 --> 00:35:07.380 All right. 00:35:07.380 --> 00:35:21.510 So what I've been saying is whenever the Hamiltonian can 00:35:21.510 --> 00:35:32.500 be expressed as a sum of individual Hamiltonians, 00:35:32.500 --> 00:35:36.790 whenever we can write the Hamiltonian this way, 00:35:36.790 --> 00:35:45.790 we can write the wave function as a product of wave 00:35:45.790 --> 00:35:59.870 functions for coordinates, xi i1, 2 N. 00:35:59.870 --> 00:36:11.660 And the energies will be the sum Ein, 00:36:11.660 --> 00:36:16.085 i equals little ei n sub i. 00:36:16.085 --> 00:36:18.740 I equals 1 to N. 00:36:18.740 --> 00:36:19.970 So this is really easy. 00:36:19.970 --> 00:36:26.780 If we have simply a Hamiltonian, which 00:36:26.780 --> 00:36:30.500 is a sum of individual particle Hamiltonians, 00:36:30.500 --> 00:36:32.330 we don't even have to stop to think. 00:36:32.330 --> 00:36:34.663 We know the wave functions and the eigenvalues. 00:36:38.310 --> 00:36:38.810 OK. 00:36:44.370 --> 00:36:56.140 Now, suppose the Hamiltonian is this plus that. 00:36:56.140 --> 00:37:00.370 So here, we have a Hamiltonian, and this is this simply the 00:37:00.370 --> 00:37:01.750 uncoupled Hamiltonian. 00:37:01.750 --> 00:37:06.340 This is what we'd like nature to be, but nature isn't so kind, 00:37:06.340 --> 00:37:08.770 and there are some coupling terms. 00:37:08.770 --> 00:37:13.960 And so we know the eigenfunctions and eigenvalues 00:37:13.960 --> 00:37:15.100 for this Hamiltonian. 00:37:15.100 --> 00:37:20.830 We call them the basis functions and the zero-order energies, 00:37:20.830 --> 00:37:22.390 and then there is this thing that 00:37:22.390 --> 00:37:24.909 couples them and leads to complications, 00:37:24.909 --> 00:37:26.200 and that's perturbation theory. 00:37:26.200 --> 00:37:27.730 We're going to do that. 00:37:27.730 --> 00:37:28.830 OK. 00:37:28.830 --> 00:37:40.850 So now, let me just say, on page nine of your notes, 00:37:40.850 --> 00:37:48.420 there's the words next time, and those are going 00:37:48.420 --> 00:37:50.495 to be replaced by you should. 00:37:54.930 --> 00:37:57.790 There's a whole bunch of things that I want you to consider, 00:37:57.790 --> 00:38:02.130 and I was planning on talking about them, 00:38:02.130 --> 00:38:03.434 but they're all pretty trivial. 00:38:03.434 --> 00:38:05.100 And so there are a whole bunch of things 00:38:05.100 --> 00:38:10.697 you should study, because I will ask you questions about them. 00:38:13.620 --> 00:38:18.000 And of greatest importance is the ability 00:38:18.000 --> 00:38:21.351 to calculate things like that. 00:38:25.050 --> 00:38:25.550 OK. 00:38:25.550 --> 00:38:29.810 Now, I'm going to give you a whole bunch of facts 00:38:29.810 --> 00:38:32.630 which I may not have derived. 00:38:32.630 --> 00:38:37.620 But you're going to live with them, 00:38:37.620 --> 00:38:39.870 and you can ask me questions. 00:38:39.870 --> 00:38:42.300 Some of these things are theorems that we can prove, 00:38:42.300 --> 00:38:45.930 but the proof of the theorem is really boring. 00:38:45.930 --> 00:38:49.330 Understanding what it is is really wonderful. 00:38:49.330 --> 00:39:01.070 So all eigenfunctions that belong 00:39:01.070 --> 00:39:05.090 to different eigenvalues-- 00:39:09.440 --> 00:39:13.930 of whatever operator we want, the Hamiltonian, 00:39:13.930 --> 00:39:17.620 some other operator-- 00:39:17.620 --> 00:39:20.155 are orthogonal. 00:39:23.340 --> 00:39:27.060 That's a fantastic simplification. 00:39:27.060 --> 00:39:29.310 So if you have two eigenfunctions 00:39:29.310 --> 00:39:36.390 of the Hamiltonian, of the position operator, anything, 00:39:36.390 --> 00:39:39.730 those eigenfunctions are orthogonal. 00:39:39.730 --> 00:39:46.450 Their integral is 0, very, very useful. 00:39:46.450 --> 00:39:53.410 Then, one of the initial postulates 00:39:53.410 --> 00:39:57.670 about quantum mechanics is this idea 00:39:57.670 --> 00:40:00.810 that the wave functions are well-behaved. 00:40:04.310 --> 00:40:08.870 Well, if I were to state it at the beginning, 00:40:08.870 --> 00:40:11.980 you wouldn't know what's well-behaved and ill-behaved, 00:40:11.980 --> 00:40:12.950 but now I can tell you. 00:40:20.240 --> 00:40:23.360 One of the things is that the wave function is continuous, 00:40:23.360 --> 00:40:25.000 no matter what the potential does. 00:40:27.680 --> 00:40:39.640 The derivative is continuous, except at an infinite barrier. 00:40:39.640 --> 00:40:42.070 So you come along, and you hit an infinite barrier, 00:40:42.070 --> 00:40:49.400 and you've already seen that with the particle in a box. 00:40:49.400 --> 00:40:52.330 The wave function is continuous at the edge of the box, 00:40:52.330 --> 00:40:54.420 but the derivative is discontinuous, 00:40:54.420 --> 00:40:58.830 and it's because it's an infinite wall. 00:40:58.830 --> 00:41:01.950 That's a pretty violent thing to make the first derivative be 00:41:01.950 --> 00:41:03.360 discontinuous. 00:41:03.360 --> 00:41:09.080 The secondary derivative is continuous, 00:41:09.080 --> 00:41:16.424 except at any sudden change in the potential. 00:41:20.140 --> 00:41:24.480 So when you're solving 1D problems, 00:41:24.480 --> 00:41:28.830 and you've got a solution that works in the various regions, 00:41:28.830 --> 00:41:31.440 and you want to connect them together, 00:41:31.440 --> 00:41:35.790 these provide some rules about the boundary conditions. 00:41:40.150 --> 00:41:46.780 So now, most real systems don't have infinite walls 00:41:46.780 --> 00:41:52.020 or infinitely sharp steps. 00:41:52.020 --> 00:41:56.164 So for calculation of physically reasonable things, 00:41:56.164 --> 00:41:58.580 wave function for the first derivative, second derivative, 00:41:58.580 --> 00:42:00.080 they are continuous. 00:42:00.080 --> 00:42:05.340 But for solving a problem, we like these steps, 00:42:05.340 --> 00:42:09.210 because then we know how to impose boundary conditions, 00:42:09.210 --> 00:42:14.420 and that gets us a much easier thing to calculate. 00:42:14.420 --> 00:42:15.020 OK. 00:42:15.020 --> 00:42:20.070 Now-- oh, that's where it went, OK-- 00:42:26.680 --> 00:42:33.200 semi classical quantum mechanics. 00:42:39.360 --> 00:42:42.420 We know that the energy classically 00:42:42.420 --> 00:42:44.646 is p squared over 2m. 00:42:44.646 --> 00:42:47.030 Right? 00:42:47.030 --> 00:42:48.980 It's 1/2 mv squared, but that's the same thing 00:42:48.980 --> 00:42:50.450 as p squared over 2m. 00:42:50.450 --> 00:42:53.340 In quantum mechanics, when we talk about Hamiltonians, 00:42:53.340 --> 00:42:58.180 the variables are x and p not x and v. 00:42:58.180 --> 00:43:01.820 So that seems like a picky thing, 00:43:01.820 --> 00:43:04.280 but it turns out to be very important. 00:43:04.280 --> 00:43:13.680 And so we can say, well, the momentum 00:43:13.680 --> 00:43:17.344 can be a function of x, classically. 00:43:24.610 --> 00:43:32.950 So I just solved this problem, and if the potential is not 00:43:32.950 --> 00:43:37.480 constant, then the momentum, classical momentum, 00:43:37.480 --> 00:43:39.000 is not constant. 00:43:39.000 --> 00:43:43.420 But we know what it is everywhere, 00:43:43.420 --> 00:43:50.880 and we also know that the wavelength is 00:43:50.880 --> 00:43:52.290 equal to h over p. 00:43:54.970 --> 00:43:59.680 So we could make a step into the unknown saying, 00:43:59.680 --> 00:44:03.280 well, the wavelength for a non-constant potential 00:44:03.280 --> 00:44:05.860 is a function of coordinate, and it's 00:44:05.860 --> 00:44:10.370 going to be equal to h over p of x. 00:44:10.370 --> 00:44:14.180 That's semi-classical quantum mechanics, everything 00:44:14.180 --> 00:44:16.230 you would possibly want. 00:44:16.230 --> 00:44:18.110 Now, for one-dimensional problems, 00:44:18.110 --> 00:44:25.960 you can solve in terms of this coordinate dependent wavelength 00:44:25.960 --> 00:44:29.900 which is related to the local momentum. 00:44:29.900 --> 00:44:34.340 And so it doesn't matter how complicated the problem is, 00:44:34.340 --> 00:44:37.610 you know that you can calculate the spatial modulation 00:44:37.610 --> 00:44:40.790 frequency, you could calculate the amplitude, 00:44:40.790 --> 00:44:45.860 is it big or small, based on these ideas 00:44:45.860 --> 00:44:50.090 of the classical momentum function. 00:44:50.090 --> 00:44:52.670 So this demonstration of my walking 00:44:52.670 --> 00:44:59.060 across the room slow, fast, slow tells you about probability. 00:44:59.060 --> 00:45:05.390 So if you use this formula, you know the node spacings, 00:45:05.390 --> 00:45:09.340 and you know the amplitudes. 00:45:09.340 --> 00:45:13.770 Now, what you don't know is where are the nodes? 00:45:13.770 --> 00:45:17.520 You know how far they are apart, but I 00:45:17.520 --> 00:45:19.230 have to be humble about this. 00:45:19.230 --> 00:45:21.030 In order to calculate where the nodes are, 00:45:21.030 --> 00:45:25.980 I have to do a little bit more in order to pin them down. 00:45:25.980 --> 00:45:28.800 But mostly, when you're trying to understand how something 00:45:28.800 --> 00:45:33.350 works, you want to know the amplitude of the envelope, 00:45:33.350 --> 00:45:36.180 and that's a probability, and so it's 00:45:36.180 --> 00:45:40.320 related to 1 over the square root of the momentum. 00:45:40.320 --> 00:45:42.540 I'm sorry, the amplitude is 1 over the square root 00:45:42.540 --> 00:45:44.371 of the momentum, and the nodes spacings, 00:45:44.371 --> 00:45:45.870 those are the things you want to do, 00:45:45.870 --> 00:45:47.775 and you want to know them immediately. 00:45:53.230 --> 00:45:56.460 And a couple other facts that I told you earlier, but I think I 00:45:56.460 --> 00:45:58.360 want to emphasize them-- 00:45:58.360 --> 00:46:08.920 the energy order, number of nodes, 00:46:08.920 --> 00:46:10.090 number of internal nodes. 00:46:14.200 --> 00:46:17.410 For 1D problems, you never skip a number of-- 00:46:17.410 --> 00:46:22.626 so you can't say, there is no wave function with 13 nodes, 00:46:22.626 --> 00:46:26.310 even if you don't like being unlucky. 00:46:26.310 --> 00:46:33.470 And it's there, and so if you want the 13th energy level, 00:46:33.470 --> 00:46:38.170 you want something with 12 nodes, 00:46:38.170 --> 00:46:40.160 and that also focuses things. 00:46:40.160 --> 00:46:42.800 So these are amazingly wonderful things, 00:46:42.800 --> 00:46:44.930 because you can get them from what 00:46:44.930 --> 00:46:48.280 you know about classical mechanics, 00:46:48.280 --> 00:46:53.750 and it's easy to embed them into a kind of half quantum 00:46:53.750 --> 00:46:56.080 mechanics. 00:46:56.080 --> 00:46:58.360 And since, I told you, this course 00:46:58.360 --> 00:47:03.340 is for use and insight, not admiration of philosophy 00:47:03.340 --> 00:47:07.160 or historical development, and this is what you want to do. 00:47:07.160 --> 00:47:09.580 You look at the problem and sketch 00:47:09.580 --> 00:47:12.460 how is the wave function going to behave 00:47:12.460 --> 00:47:15.010 and perhaps how a particular thing 00:47:15.010 --> 00:47:16.630 at some place in the potential is 00:47:16.630 --> 00:47:19.840 going to affect the energy levels or any other observable 00:47:19.840 --> 00:47:21.080 property. 00:47:21.080 --> 00:47:25.660 And so the cartoons are really your guide to getting things 00:47:25.660 --> 00:47:28.770 right, but you really have to invest 00:47:28.770 --> 00:47:35.491 in developing the sense of how to build these cartoons. 00:47:35.491 --> 00:47:35.990 OK. 00:47:35.990 --> 00:47:37.830 I'm finished early again. 00:47:37.830 --> 00:47:39.260 Does anybody have any questions? 00:47:42.020 --> 00:47:42.520 OK. 00:47:42.520 --> 00:47:46.190 We start the harmonic oscillator next time. 00:47:46.190 --> 00:47:48.870 OK, so I can say a couple of things. 00:47:48.870 --> 00:47:52.320 The wave functions, the solution to the 1D 00:47:52.320 --> 00:47:56.570 box and the free particle, they're really simple. 00:47:56.570 --> 00:47:58.280 The solution to the harmonic oscillator 00:47:58.280 --> 00:48:01.740 involves a complicated differential equation 00:48:01.740 --> 00:48:04.385 which the mathematicians have solved and worked 00:48:04.385 --> 00:48:07.210 out all the properties. 00:48:07.210 --> 00:48:12.610 But there is a really important simplification 00:48:12.610 --> 00:48:17.590 that enables us to proceed with even greater velocity 00:48:17.590 --> 00:48:19.120 in the harmonic oscillator than we 00:48:19.120 --> 00:48:20.920 would in a particle in a box. 00:48:20.920 --> 00:48:26.320 And they are these things called a and a 00:48:26.320 --> 00:48:30.370 dagger, creation and annihilation operators, where 00:48:30.370 --> 00:48:37.450 when we operate on psi with this creation operator, 00:48:37.450 --> 00:48:48.460 it converts it to square root of v plus 1, psi v plus 1 further. 00:48:48.460 --> 00:48:51.520 These things, we don't ever have to do an integral. 00:48:51.520 --> 00:48:53.530 Once you're in harmonic oscillator land, 00:48:53.530 --> 00:49:00.490 everything you need comes from these wonderful operators. 00:49:00.490 --> 00:49:02.650 And so even though the differential equation 00:49:02.650 --> 00:49:07.070 is a little bit scary for chemists, 00:49:07.070 --> 00:49:10.200 these things make everything trivial. 00:49:10.200 --> 00:49:14.460 And so we use the harmonic oscillator, 00:49:14.460 --> 00:49:16.910 and the particle in a box to illustrate 00:49:16.910 --> 00:49:19.440 time-dependent quantum mechanics. 00:49:19.440 --> 00:49:24.840 They each have their own special advantages for simplifications, 00:49:24.840 --> 00:49:30.390 but it's wonderful, because we can use something we barely 00:49:30.390 --> 00:49:32.550 understand for the first time. 00:49:32.550 --> 00:49:34.590 And actually reach that level of, yeah, I 00:49:34.590 --> 00:49:38.030 can understand macroscopic behaviors too 00:49:38.030 --> 00:49:42.180 and how they relate to quantum mechanical behavior 00:49:42.180 --> 00:49:43.760 of simple systems. 00:49:43.760 --> 00:49:45.115 OK, so that's where we're going. 00:49:45.115 --> 00:49:46.823 We're going to have two or three lectures 00:49:46.823 --> 00:49:49.460 on harmonic oscillator.